Berezin transforms attached to Landau levels on the complex projective space CPn

In this paper, we construct coherent states for each generalized Bergman space on the n-dimensional complex projective space in order to apply a coherent states quantization method. Doing so allows to define the Berezin transform for these spaces. In particular, we provide a variational formula for this transform by means of the Fubini-Study Laplace operator which reduces when n = 1 and for the lowest spherical Landau level to the well-known formula previously given by Berezin himself.